Lafforgue has proposed a new approach to the principle of functoriality in a test case, namely, the case of automorphic induction from an idele class character of a quadratic extension. For technical reasons, he considers only the case of function fields and assumes the data is unramified. In this paper, we show that his method applies without these restrictions. The ground field is a number field or a function field and the data may be ramified.

The aim of these notes is to generalize Laumon’s construction [20] of automorphic sheaves
corresponding to local systems on a smooth, projective curve C to the case of local
systems with indecomposable unipotent ramification at a finite set of points. To this end
we need an extension of the notion of parabolic structure on vector bundles to coherent
sheaves. Once we have defined this, a lot of arguments from the article “ On the
geometric Langlands conjecture” by Frenkel, Gaitsgory and Vilonen [11]...

We prove that the global geometric theta-lifting functor for the dual pair (H,G) is compatible with the Whittaker functors, where (H,G) is one of the pairs (S𝕆2n,𝕊p2n), (𝕊p2n,S𝕆2n+2) or (𝔾Ln,𝔾Ln+1). That is, the composition of the theta-lifting functor from H to G with the Whittaker functor for G is isomorphic to the Whittaker functor for H.

Let X be a smooth projective curve over an algebraically closed field of characteristic &gt;2. Consider the dual pair H= SO 2m,G= Sp 2n over X with H split. Write Bun G and Bun H for the stacks of G-torsors and H-torsors on X. The theta-kernel Aut G,H on Bun G× Bun H yields theta-lifting functors FG:D( Bun H)→D( Bun G) and FH:D( Bun G)→D( Bun H) between the corresponding derived categories. We describe the relation of these functors with Hecke operators.
In two particular cases these functors realize the geometric Langlands functoriality for the above pair (in the non ramified case)....